Analysis of Profit and Marked Price in Discounted Sales
Understanding the relationship between the marked price, cost price, and the profit earned is crucial for any retail business. This article explores a common scenario where a shopkeeper provides a discount on the marked price, yet manages to maintain a specific profit margin.
Introduction to the Scenario
Consider a shopkeeper who marks his goods in such a way that after allowing a 25% discount on the marked price (MP), he still makes a profit of 50% on the cost price (CP). This article aims to determine the ratio of the cost price to the marked price and the relative increase in the marked price compared to the cost price.
Calculating the Ratio of Cost Price to Marked Price
We start with the following relationship:
100/100C - 25/100M 50/100C
This can be simplified to:
3/4M 3/2C
Rearranging the terms, we get:
M 2C
The ratio of the cost price to the marked price is therefore 1:2.
Example Calculation
Let's consider an example where the cost price is Rs 100 and the profit is 50%. The selling price (SP) after a 25% discount on the marked price (MP) should still achieve this profit margin.
Step-by-Step Solution
Step 1: Determine the selling price (SP) after a 25% discount.
The selling price is the cost price plus 50% profit:
SP 100 50 150
Step 2: Apply the 25% discount on the marked price to achieve the selling price.
SP MP - 25% of MP
150 MP - 0.25MP
150 0.75MP
MP 150 / 0.75 200
Step 3: Calculate the ratio of the cost price to the marked price.
The ratio of CP to MP is 100 / 200 1:2.
Percentage Increase in Marked Price
To find out what percent more the marked price is compared to the cost price, we use the following equations.
If the cost price (CP) is x and the selling price (SP) is 1.4x (since the shopkeeper makes a 40% profit), and the selling price is also 0.75 times the marked price (MP), we can set up the following equation:
1.4x 0.75MP
Rearranging to solve for MP:
MP (1.4x) / 0.75 (14x) / 7.5 (28x) / 15
Now, to find the percentage increase of MP compared to CP:
Percentage increase ((MP - CP) / CP) * 100
Substituting MP (28x / 15) and CP x:
Percentage increase ((28x / 15 - x) / x) * 100
Percentage increase ((28x - 15x) / 15) * 100
Percentage increase (13x / 15) * 100
Percentage increase 86.67%
Therefore, the marked price is approximately 86.67% more than the cost price.
Finding Cost Price with Given Data
Another scenario is where the marked price is Rs 10, and after a 25% discount, the sale price is Rs 9. We need to find the cost price where a profit of 26% is made.
Step 1: Calculate the cost price.
CP 100 / 126 * 9 71.4286
Step 2: Calculate the percentage increase.
28.5714 / 71.4286 * 100 40%
Thus, the marked price is approximately 40% more than the cost price in this scenario.
Conclusion
This article has provided a detailed analysis of the relationship between marked price, cost price, and the profit an individual can make after offering a discount. Understanding these concepts is crucial for businesses aiming to optimize their pricing strategies for maximum profitability.
Key Takeaways:
The ratio of the cost price to the marked price is determined by the profit margin and discount rate. The marked price is typically higher than the cost price, and the increase in percentage can be calculated for any given profit margin and discount. Maintaining a clear understanding of these principles helps businesses make informed decisions in pricing their goods.